Question: Simplify the following expression and state the condition under which the simplification is valid. $z = \dfrac{-10n^2 + 70n + 300}{-8n^3 + 32n^2 + 168n}$
Answer: First factor out the greatest common factors in the numerator and in the denominator. $ z = \dfrac {-10(n^2 - 7n - 30)} {-8n(n^2 - 4n - 21)} $ $ z = \dfrac{10}{8n} \cdot \dfrac{n^2 - 7n - 30}{n^2 - 4n - 21} $ Simplify: $ z = \dfrac{5}{4n} \cdot \dfrac{n^2 - 7n - 30}{n^2 - 4n - 21}$ Next factor the numerator and denominator. $ z = \dfrac{5}{4n} \cdot \dfrac{(n + 3)(n - 10)}{(n + 3)(n - 7)}$ Assuming $n \neq -3$ , we can cancel the $n + 3$ $ z = \dfrac{5}{4n} \cdot \dfrac{n - 10}{n - 7}$ Therefore: $ z = \dfrac{ 5(n - 10)}{ 4n(n - 7)}$, $n \neq -3$